We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-<i>O</i>(1/<i>n</i>). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node <i>u</i> and any radius <i>r</i> the number of nodes within distance 2<i>r</i> from <i>u</i> is at most a constant times the number of nodes within distance <i>r</i>, our augmentation process proceeds with high probability in <i>O</i>(log <i>n</i> log <i>D</i>) communication rounds, with <i>O</i>(log <i>n</i> log <i>D</i>) messages of size <i>O</i>(log <i>n</i>) bits sent per node and requiring only <i>O</i>(log <i>n</i> log <i>D</i>) bit space in each node, where <i>n</i> is the number of nodes, and <i>D</i> the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most <i>O</i>(log<sup>2</sup> <i>n</i> log<sup>2</sup> <i>D</i>) with probability 1 - <i>O</i>(1/<i>n</i>), and of expected length <i>O</i>(log <i>n</i> log<sup>2</sup> <i>D</i>). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinberg's model as a plausible explanation for the small world phenomenon in large real interaction networks.
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